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I've been computing the angles of a triangle with sides a = 17, b = 6 and c = 15 using the law of cosines to find the first angle and then the law of sines to find the other 2. I follow the convention of naming the angles opposite these sides A, B and C respectively. Here are my results:

$ C = \arccos( \frac {6^2+17^2-15^2}{2(6)(17)}) = 60.647$ degrees to 3 d.p.

$ B = \arcsin( \frac {6 \sin C}{15}) = 20.405$ degrees to 3 d.p.

$ A = \arcsin( \frac {17 \sin B}{6}) = 81.051$ degrees to 3 d.p.

Clearly, adding these should give $180$ degrees, but it gives 162 degrees to 3 s.f. Assuming I haven't made any mistakes, the error seems quite high and I'm just wondering if anyone knows why this is? It seems high enough to challenge the validity of the laws.

George Tomlinson
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    Maybe it's because of the ambiguous case, that arises when using the Law of sines. Why don't you do the Law of Cosines two times and subtract from 180° to find the third angle? No need for Law of Sines here. – imranfat Oct 15 '13 at 19:07
  • Because if I do that, it's not testing the accuracy of the law. +1 because info and question useful. – George Tomlinson Oct 15 '13 at 19:14
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    The laws are acurate and so is your calculator for the trig terms, but the ambiguous case is the issue – imranfat Oct 15 '13 at 19:16
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    Note that the error is about $18$ degrees, which is just the difference between $81$ degrees and $180-81=99$ degrees. Those angles have the same sine. You are picking the wrong one. – Ross Millikan Oct 15 '13 at 19:19
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    Hey Ross, congrats BTW on your repuation points. In essence, isn't picking the "wrong" one inherently related to the ambiguous case due to sinx = sin(180-x) ? – imranfat Oct 15 '13 at 19:21
  • I can see it's sensible for the calculator just to choose the first angle greater than or equal to 0 that works. +1 for useful info. @Ross Millikan – George Tomlinson Oct 15 '13 at 19:22
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    @imranfat: The arcsin function is defined (so as to be single valued) to return values between $-90$ and $+90$ degrees. The error was made in going from $\sin A= ()$ to $A=\arcsin ()$. Those are not equivalent. It is the same as going from $x^2=2$ to $x=\sqrt 2$ and missing the $\pm$ sign. The calculator is useful and returned the correct answer to the question it was asked. – Ross Millikan Oct 15 '13 at 19:27
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    Regardless of the ambiguous case, remember that $\sin(\theta^\circ) = \sin(180^\circ - \theta^\circ)$. So, if you compute $\arcsin u = \theta^\circ$, then you have to find out if $\theta^\circ$ or $180^\circ - \theta^\circ$ is the angle that you are really looking for. Arccosines don't have that problem. – Steven Alexis Gregory Mar 26 '16 at 08:26

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OK, I did the Law of Cosines 3 times and came up with 60.647 , 20.404 and 98.949 respectively for angles A, B and C. Remember, the Law of Cosines does not have an ambiguous case, unlike the Law of Sines. I suspect (without further investigating) that his may be the culprit. My advice: Always use the Law of Cosines whenever you can. In this case, when all sides are known, clearly a case for Law of Cosines

imranfat
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Switching from Law of Cosines to Law of Sines may introduce the ambiguous case and create extraneous solutions, so it is better to stick with Law of Cosines as much as possible. If you do change to Law of Sines, you can test your results by substituting ALL of your sides and angles into the proportion. If you do not obtain equivalent results, you have the extraneous solution and will need to rework the problem using the supplement of the angle you initially obtained. The link connects to Google Slides I prepared for my students. Testing solutions using Law of Sines

N. Compton
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Remember that $$sin(180-θ)=sinθ$$

$sin(180-81.051)=sin(98.949)=0.987$

$60.647+20.405+81.051=162.103$

$60.647+20.405+98.949=180.001$

The correct angle should be 98.949

In graph perspective, positive cosine means acute angle(Q1) while negative cosine means obtuse angle(Q2). But with sine we have to test if the angle is in Q1 or Q2 since it has the same sine values of 0.987. Hence, we calculate which degree of the two adds up to 180. Therefore, as mentioned by others, it is advisable to use law of cosines.

momokjaaaaa
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